Practice Test 34 - Algebra 2

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. What is the coefficient of $x^4 y^2$ in the expansion of $(x + y)^6?$
2. A small town has a population of 15,000 and grows by 4% each year. Model the population with an exponential equation and determine when the population will reach 18,000. Let $t$ = number of years.
3. A ball is thrown upward from a platform. Its height above the ground (in meters) after $t$ seconds is given by $h(t)=-5t^2+20t+2$. Graph the equation with appropriate axes labels and a reasonable scale for $0\le t\le 5$.
4. For $x^2-6x+10=0$, first note the discriminant is $b^2-4ac$. What type of solutions does this equation have, and what are the solutions? (Write complex solutions in $a\pm bi$ form.)
5. Find the modulus $r$ and a correct argument $\theta$ (in degrees) for $z=-2+2i$, then write $z$ in polar form $r(\cos\theta+i\sin\theta)$. Use $r=\sqrt{a^2+b^2}$ and $\theta=\arctan(b/a)$ with quadrant adjustment. (Argument is measured counterclockwise from the positive real axis.)
6. Let $z=1-2i$ and $w=3+i$. Compute $\overline{zw}$. (You may use $\overline{zw}=\overline{z}\,\overline{w}$.)
7. The complex number $w = 2\left(\cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}\right)$ can be written in rectangular form as $a + bi$. What is the value of $a - b$?
8. A savings account is opened with $2500 and earns $4.5\%$ interest compounded annually. Create an exponential model for the balance $A(t)$ after $t$ years, then determine how long it will take for the account to reach $4000. Round your time to the nearest tenth of a year and interpret your result in context.
9. A company's weekly profit (in thousands of dollars) is modeled by $P(x)=(x-1)(x-5)$, where $x$ is the number of thousands of items sold that week. On what interval is the profit positive?
10. What are the period and amplitude of the trigonometric function $q(x)=-4\cos\left(\dfrac{x}{2}\right)$? (Use the parent $y=\cos(x)$.)
11. A table compares a linear function $L(x)=20x$, a quadratic $Q(x)=x^2$, a polynomial $P(x)=x^4$, and an exponential $E(x)=2^x$. Based on the values shown, which statement is most accurate about what happens for large $x$?
12. Write matrices $A$, $X$, and $B$ for the system (use variable order $x, y$): $$\begin{cases} -2x+7y=4\\ 5x+y=10 \end{cases}$$ Which choice correctly gives $AX=B$?
13. Points $z_1=-1+4i$ and $z_2=5-4i$ lie on the complex plane. Using the geometric interpretation that $|z_1-z_2|$ is the length of the vector from $z_2$ to $z_1$ and equals $\sqrt{(a-c)^2+(b-d)^2}$, what is the distance between them?
14. Use the finite geometric series formula $S_n=\dfrac{a(1-r^n)}{1-r}$ (for $r\ne 1$) to find the sum $2+2(1.1)+2(1.1)^2+\cdots+2(1.1)^9.$
15. Use the table to approximate the solution to $f(x)=g(x)$, where $f(x)=x^3$ and $g(x)=2x+5$. Choose the best estimate for the solution to the nearest tenth. Table:

    $x$	$f(x)=x^3$	$g(x)=2x+5$
    1.5	3.375	8.0
    1.8	5.832	8.6
    1.9	6.859	8.8
    2.0	8.000	9.0
    2.1	9.261	9.2
    2.2	10.648	9.4

16. In the expression $3(2x+1)^2$, what is the coefficient of $(2x+1)^2$?
17. Sketch $f(x)=(x+4)(x+1)(x-2)^2$ showing all zeros, their multiplicities, and the correct end behavior. Which description matches?
18. Function $f$ is given algebraically by $f(x)=\dfrac{x+4}{x-1}$. Function $g$ is described verbally as: "$g$ is a rational function with a vertical asymptote at $x=-3$." Which function has a vertical asymptote farther from the origin on the x-axis?
19. Solve by completing the square: $x^2+12x+20=0$ Write it as $(x-p)^2=q$ and solve.
20. Two functions are shown: - Polynomial: $h(x)=x^4$ - Exponential: $p(x)=2^x$ Which statement best describes what happens after the crossover point where $2^x$ becomes larger than $x^4$?​

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